An uncertain currency model with floating interest rates

Soft Comput 17 1:6739 6754 DOI 1.17/s5-16-4-9 MTHODOLOGIS AND APPLICATION An uncertain currency model with floating interest rates Xiao Wang 1 Yufu Ning 1 Published online: June 16 Springer-Verlag Berlin Heidelberg 16 Abstract Considering the uncertain fluctuations in the financial market from time to time, we propose a currency model with floating interest rates within the framework of uncertainty theory. Different from the classical stochastic currency models, this paper is assumed that the domestic interest rate, the foreign interest rate the exchange rate follow uncertain differential equations. After that, the pricing formulas of uropean American currency options are derived. The simulation experiments presented in this paper illustrate the performance of the proposed model, the relationship between the option pricing formulas all relevant paramete is analyzed. Keywords Currency model Option pricing formula Floating interest rate Uncertain differential equation 1 Introduction With the fast growing of the trading volume in the foreign exchange market, the trading of currency option has been increasing. A currency option is a contract, which gives the owner the right, but not the obligation, to buy or sell the indicated amount of foreign currency at a specified price on a fixed date uropean option or within a specified period of time American option. As we know, the currency option can be used as a tool for corporations or individuals to avoid risks Communicated by V. Loia. B Xiao Wang wangxiao1987115@163.com 1 School of Information ngineering, Shong Youth Univeity of Political Science, Jinan 513, China obtain profits. Hence, an appropriate pricing formula for currency option is becoming extremely significant. Within the frame work of probability theory, the existing academic literatures on the pricing of foreign currency options could be divided into two categories. At fit, both domestic foreign interest rates are assumed to be constant, whereas the spot exchange rate is assumed to follow a stochastic differential equation. In 1983, Garman Kohlhagen 1983 presented G K model for pricing the uropean currency option under the assumption that the dynamics of the spot exchange rate is governed by a geometric Brownian motion with constant drift volatility. The pricing formula is one of the veions of the Black Scholes option pricing formula Black Scholes 1973. It is documented in the subsequent literatures that the geometric Brownian motion assumption for the dynamics of the spot exchange rate is not realistic. Since then, in order to avoid this drawback, many methodologies for pricing currency options have been proposed by using modifications of the G K model, such as Bollen Rasiel 3, Carr Wu 7, Hilliard et al. 1991, Melino Turnbull 199, Sarwar Krehbiel, Sun 13, Swishchuk 14 Xiao et al. 1 among othe. The second class of models for pricing foreign currency options incorporated stochastic interest rates. Grabbe 1983 developed Grabbe model for currency option pricing when the interest rates were assumed to be stochastic, pointed out that both American call put option prices were larger than their uropean counterparts. From then on, numerous extensions of Grabbe model for the currency option pricing have been proposed, such as Amin Jarrow 1991, Heston 1993, van Haastrecht Pelsser 11 Xu 6. In the literature mentioned above, all the currency models interest rate models are built by probability theory. It is well known that a premise of applying probability the-

674 X. Wang, Y. Ning ory is that the obtained samples are adequate available to estimate a probability distribution. However, indeterminacy unexpected events are inevitable in the real world, such as wa, natural disaste government intervention, which bring uncertainties into in the financial market. Owing to just few available statistical data or even no samples, it is inappropriate to adhere to the assumption that the exchange rate interest rates follow stochastic differential equations in this case. As the result, we have to invite some domain experts to evaluate their belief degree that each event will occur. This provided a motivation for Liu 7 to establish an uncertainty theory, which can deal with the belief degree mathematically. Based on normality, duality, subadditivity product axioms, uncertainty theory has become a branch of axiomatic mathematics to model human uncertain behavior. To describe dynamic uncertain system, Liu 8 proposed the concept of uncertain process. As a special type of uncertain process, Liu 9 presented canonical Liu process which could be considered as a counterpart of Brownian motion. Besides, based on the canonical Liu process, Liu introduced uncertain integral Liu 9 uncertain differential equation Liu 8. In the complicated financial market, the changes which arise from political policies, social events, wa, natural disaste many other unknown facto will more or less effect the variation trend of exchange rate interest rates in the near future, it is almost impossible to obtain an exact estimation computed by probability theory. In this case, under the assumption that the interest rate follows a uncertain differential equation, Chen Gao 13 fit proposed three different uncertain interest rate models which are the counterparts of the Ho Lee model 1986, Vasicek model 1977 Cox Ingeoll Ross model 1985, respectively. After that, Jiao Yao 15 solved the proposed another uncertain interest rate model using Runge Kutta method. With the use of uncertain fractional differential equation, Zhu 15 recently presented another uncertain interest rate model obtained the zero-coupon bond price. Based on the assumption that the exchange rate follows an uncertain differential equation, Liu et al. 15 fit proposed an uncertain currency model for the foreign exchange market in 15. However, both the domestic foreign interest rates are taken as constants in the Liu Chen Ralescuet model. In fact, due to the fluctuation of financial markets from time to time, the interest rate is changing instead of a constant. In this paper, we further study currency option pricing problem with the framework of uncertainty theory propose a new uncertain currency model with floating interest rates. The rest of the paper is organized as follows. In Sect., uncertain differential equation, uncertain currency model uncertain interest rate model are briefly introduced for the completeness of this research. Section 3 introduces the new currency model with floating interest rates. uropean American currency option pricing formulas are derived in Sect. 4. And some numerical experiments are presented to show the performance of the new currency model in Sect. 5. After that, Sect. 6 derives the currency option pricing formulas under a more general currency model. Finally, we make a brief conclusion in Sect. 7. For convenience, some notations used in the later sections are introduced as follows: 1 α: the invee uncertainty distribution of stard normal uncertain variable, i.e., 3 1 α π ln α 1 α ; r t : the domestic interest rate at time t; f t : the foreign interest rate at time t; Z t : the spot domestic currency price of a unit of foreign exchange at time t; μ: the log-drift of the spot currency price Z t ; σ : the log-diffusion of the spot currency price Z t ; K : the strike price; T : the maturity date; C t : a canonical { Liu process; x, if x x : x, if x <. Preliminaries In this section, we fit introduce some basic concepts theorems about uncertain differential equation based on uncertain variable uncertain process. For more details about them, see Appendices 1. Then, uncertain currency model uncertain interest rate model are recalled..1 Uncertain differential equation Definition.1 Liu 8 Suppose C t is a canonical Liu process, f g are two functions. Given an initial value X, dx t f t, X t dt gt, X t dc t 1 is called an uncertain differential equation with an initial value X. In essence, uncertain differential equation is a type of differential equation driven by a canonical Liu process, while stochastic differential equation is driven by a Wiener process. Uncertain differential equation 1 is equivalent to an integral equation X t X f s, X s ds gs, X s dc s,

An uncertain currency model with floating interest rates 6741 a solution of 1 is just an uncertain process X t satisfying the integral equation. Definition. Yao Chen 13 Theα-path <α< 1 of uncertain differential equation 1 with initial value X is a deterministic function Xt α with respect to t that solves the corresponding ordinary differential equation dx α t f t, X α t dt gt, X α t 1 αdt, X α X. Theorem.1 Yao Chen 13 Assume that X t X α t are the solution α-path of the uncertain differential equation 1, then M{X t Xt α, t} α, M{X t > Xt α, t} 1 α, where M is an uncertain measure see Appendix 1. Theorem. Yao 15 Let X t Y t be the solutions of uncertain differential equations dx t f 1 t, X t dt g 1 t, X t dc 1t dy t f t, Y t dt g t, Y t dc t, 3 respectively, where C 1t C t are two independent canonical Liu processes. Then, for any positive numbe T K, we have exp exp exp exp t T t T ] X s ds Y T K X s ds exp t T exp Xs 1α ds YT α K, ] K Y T Xs α ds K YT α, ] X s ds Y t K exp t T exp Xs 1α ds Yt α K ] X s ds K Y t X α s ds K Y α t, where Xt α Yt α are the α-path of uncertain differential equations 3, respectively.. Uncertain currency model Uncertain differential equations play a important role in solving the financial problems. In 15, Liu et al. 15 assumed that the exchange rate Z t follows an uncertain differential equation, proposed an uncertain currency model as below: dx t rx t dt dy t fy t dt 4 dz t μz t dt σ Z t dc t which is the uncertain counterpart of the G K model Garman Kohlhagen 1983, where X t represents the riskless domestic currency with domestic interest rate r, Y t represents the riskless foreign currency with foreign interest rate f, Z t represents the exchange rate that is domestic currency price of one unit of foreign currency at time t. Model 4 with fixed interest rates is usually called as Liu Chen Ralescu model. Assume that a uropean currency call option subjected to the Liu Chen Ralescu model has a strike price K a maturity date T. Liu et al. 15 derived the pricing formula of uropean currency call option as follows: P exp rt exp ft Z exp μt σ T 1 α K Z.3 Uncertain interest rate model K exp μt σ T 1 α. In 13, Chen Gao 13 proposed a mean-reverting uncertain interest rate model as follows: dr t m ar t dt σ dc t 5 where σ is the constant variance rate, a represents the rate of adjustment of r t m/a is the long run average value of r t. This model can be seen as the uncertain counterpart of the Vasicek model 1977 is often referred to as Chen Gao interest rate model. 3 Uncertain currency model with floating interest rates In this section, we make some improvements referring to Liu Chen Ralescu model. Considering the vague fluctuation of financial markets from time to time, the interest rate is usually changing. Hence, it is impractical to assume that the domestic foreign interest rates are constants. In what follows, assuming that the domestic interest rate, foreign interest

674 X. Wang, Y. Ning rate the exchange rate follow uncertain differential equations, we then propose a new currency model as below, dr t m 1 r t dt σ 1 dc 1t d f t m f t dt σ dc t 6 dz t μz t dt σ Z t dc 3t where the constant σ 1 is the diffusion of the domestic interest rate r t ; the constant σ is the diffusion of the foreign interest rate f t ; m 1, m, are the constant paramete; C 1t, C t C 3t are independent canonical Liu processes. According to Theorem.1, it is easy to verify that the α-path of the domestic interest rate r t is rt α m 1 exp t r m 1 σ 1 1 exp t 1 α, 7 the α-path of the foreign interest rate f t is ft α m exp t f m σ 1 exp t 1 α 8 the α-path of exchange rate Z t is Z α t Z exp μt σ t 1 α. 9 4 Pricing formulas of currency option In this section, we study the uropean American currency option present the pricing formulas of uropean American currency option under model 6. For the sake of simplifying the notation, we pose that the current time is. The case of a generic current time t can be extended easily by following the same method. 4.1 uropean currency option pricing We shall fit consider the pricing of a uropean call option on the currency with a strike price K a maturity date T. More specifically, a uropean currency call option is a contract that gives the holder the right without the obligation to buy one unit of foreign currency at a maturity date T for K units of domestic currency, where K is commonly called a strike price. Now, we start to derive an integral representation for a uropean call option with maturity at time T strike price K under model 6. Assume that the price of this contract is C in domestic currency. Then, the investor pays C for buying this contract at time. At the maturity date T, if the current exchange rate Z T is not less than the strike price K, then the investor exercises the right receives Z T K in domestic currency; if the current exchange rate Z T is less than the strike price K, then the investor gives up to exercise the right. So the expected profit of the investor at time is C ] Z T K exp. On the other h, the bank receives C for selling the contract at time pays 1 K/Z T in foreign currency at the maturity date T. Hence, the expected profit of the bank at time is C Z 1 K/Z T exp ] f s ds. In order to make a fair deal, the investor the bank should have the same expected profit, i.e., C C Z Z T K exp ] 1 K/Z T exp ] f s ds. Thus, the uropean currency call option price is given by the definition below. Definition 4.1 Assume a uropean currency call option has a strike price K a maturity date T. Then, the uropean currency call option price C is C 1 Z T K exp Z 1 K/Z T exp ] ] f s ds. Next, we provide an integral representation for the pricing formula of the uropean currency call option. Theorem 4.1 Assume a uropean currency call option under model 6 has a strike price K a maturity date T. Then, the uropean currency call option price is C 1 Z Z α T K exp 1 K/Z α T exp 1α ds 1α ds

An uncertain currency model with floating interest rates 6743 where ZT α K 1 K/ZT α 1 exp exp exp exp Z exp 1α ds m 1T σ 1 1 1 α 1α ds m T σ 1 1 α μt σ T 1 α K, K Z exp μt σ T 1 α, r m 1 exp T 1 T 1 exp T 1 f m exp T 1 T 1 exp T 1. Proof It follows from Theorem. that Z T K exp Z α T K exp ] 1 K/Z T exp 1 K/Z α T exp 1α ds ] f s ds 1α ds. Thus, the pricing formula of uropean currency call option follows from Definition 4.1 formulas 7, 8 9 immediately. We have derived the pricing formula of uropean currency call option in the previous part, then, we immediately discuss the properties of this pricing formula. Theorem 4. Let C be the uropean currency call option price under model 6. Then, 1 C is a decreasing function of m 1 ; C is a decreasing function of m ; 3 C is an increasing function of μ; 4 C is a decreasing function of r ; 5 C is a decreasing function of f ; 6 C is an increasing function of Z ; 7 C is a decreasing function of K. Proof By Theorem 4.1, the pricing formula of uropean currency call option can be expressed as C 1 Z expμt σ T 1 α K exp m r 1T m 1 exp T 1 σ 1 1 1 α T 1 exp T 1 1 K Z expμt σ T 1 α exp m f T m exp T 1 σ 1 1 α T 1 exp T 1. 1 Since T exp T 1 >, T >, the function exp m 1T m 1 a1 exp T 1 exp m 1 a1 T exp T 1 is decreasing with respect to m 1 the uropean currency call option price C is decreasing with respect to the parameter m 1. Since T exp T 1 >, T >, the function exp m T m a exp T 1 exp m a T exp T 1 is decreasing with respect to m the uropean currency call option price C is decreasing with respect to the parameter m. 3 Since the functions Z expμt σ T 1 α K Z K/ expμt σ T 1 α K, Z, T > are increasing with respect to μ, the uropean currency call

6744 X. Wang, Y. Ning option price C is increasing with respect to the log-drift μ of exchange rate. 4 Since the function expr is decreasing with respect to r, the uropean currency call option price C is decreasing with respect to the initial domestic interest rate r. 5 Since the function exp f is decreasing with respect to f, the uropean currency call option price C is decreasing with respect to the initial foreign interest rate f. 6 Since the functions Z expμt σ T 1 α K Z K/ expμt σ T 1 α are increasing with respect to Z, the uropean currency call option price C is decreasing with respect to the initial exchange rate Z. 7 Obviously, the functions Z expμt σ T 1 α K Z K/ expμt σ T 1 α are decreasing with respect to K. Thus, the uropean currency call option price C is decreasing with respect to the strike price K. By following the same method as uropean currency call option, we obtain the following definition, formula properties of uropean currency put option pricing with strike price K maturity at time T. exp m r 1T m 1 exp T 1 σ 1 1 α exp exp f α s ds m T σ 1 α T 1 exp T 1 f m exp T 1 T 1 exp T 1. Proof It follows from Theorem. that K Z T exp Z α T K exp ] r α s ds Definition 4. Assume a uropean currency put option has a strike price K a maturity date T. Then, the uropean currency put option price P is P 1 K Z T exp Z K/Z T 1 exp ] ] f s ds. Theorem 4.3 Assume a uropean currency put option under model 6 has a strike price K a maturity date T. Then, the uropean currency put option price is P 1 Z K Z α T exp K/Z α T 1 exp r α s ds f α s ds where K ZT α K Z expμt σ T α 1, K/ZT α K 1 Z expμt σ T 1 α 1, exp α ds K/Z T 1 exp K/Z α T 1 exp ] f s ds f α s ds. Thus, the uropean currency put option pricing formula follows from Definition 4. formulas 7, 8 9 immediately. Theorem 4.4 Let P be the uropean currency put option price under model 6. Then, 1 P is a decreasing function of m 1 ; P is a decreasing function of m ; 3 P is a decreasing function of μ; 4 P is a decreasing function of r ; 5 P is a decreasing function of f ; 6 P is a decreasing function of Z ; 7 P is an increasing function of K. Proof By Theorem 4.3, the pricing formula of uropean currency put option can be written as P 1 K Z expμt σ T 1 α

An uncertain currency model with floating interest rates 6745 exp m r 1T m 1 exp T 1 σ 1 1 α T 1 exp T 1 1 K expμt σ T 1 α Z exp m f T m exp T 1 σ 1 α T 1 exp T 1. 1 Since T exp T 1 >, T >, the function exp m 1T m 1 a1 exp T 1 exp m 1 a1 T exp T 1 is decreasing with respect to m 1 the uropean currency put option price P is decreasing with respect to the parameter m 1. Since T exp T 1 >, T >, the function exp m T m a exp T 1 exp m a T exp T 1 is decreasing with respect to m the uropean currency put option price P is decreasing with respect to the parameter m. 3 Since the functions K Z expμt 3σ T π ln 1α α K/ expμt 3σ T π ln 1α α Z K, Z, T > are decreasing with respect to μ, the uropean currency put option price P is decreasing with respect to the log-drift μ of exchange rate. 4 Since the function expr is decreasing with respect to r, the uropean currency put option price P is decreasing with respect to the initial domestic interest rate r. 5 Since the function exp f is decreasing with respect to f, the uropean currency put option price P is decreasing with respect to the initial foreign interest rate f. 6 Since the functions K Z expμt K/ expμt 3σ T π ln α 1α 3σ T π ln α 1α Z are decreasing with respect to Z, the uropean currency put option price P is decreasing with respect to the initial exchange rate Z. 7 Obviously, the functions K Z expμt 3σ T π ln 1α α K/ expμt 3σ T π ln 1α α Z are increasing with respect to K. Therefore, the uropean currency put option price P is decreasing with respect to the strike price K. 4. American currency option pricing In this subsection, we shall fit consider the uncertain approach to price the American currency call option. Different from the uropean currency call option, an American currency call option is a contract that gives the holder the right without the obligation to buy one unit of foreign currency with a strike price K at any time prior to a maturity date T. In what follows, we derive an integral representation for an American currency call option with maturity at time T strike price K under model 6. Obviously, the best choice for the investor is to carry out at the reme value Z t K exp. t T Assume that C A denotes the price of this contract. On the one h, the expected return of the investor at time is C A Z t K exp ]. t T On the other h, the expected return of the bank at time is C A Z 1 K/Z t exp f s ds ]. t T In order to make a fair deal, the investor the bank should have the same expected return, i.e., C A C A t T ] Z t K exp Z 1 K/Z t exp t T f s ds ]. Thus, the American call currency option price is given by the definition below. Definition 4.3 Assume an American currency call option has a strike price K a maturity date T. Then, the Amer-

6746 X. Wang, Y. Ning ican currency call option price is C A 1 ] Z t K exp t T 1 Z 1 K/Z t exp f s ds ]. t T Theorem 4.5 Assume an American currency call option under model 6 has a strike price K a maturity date T. Then, the American currency call option price is C A 1 where Z t T Zt α K exp t T Zt α K 1 K/Zt α 1 exp exp Z exp 1α ds m 1t σ 1 1 1 α exp exp 1α ds m t σ 1 1 α 1 K/Zt α exp 1α ds 1α ds μt σ t 1 α K, K Z exp μt σ t 1 α, r m 1 exp t 1 t 1 exp t 1 f m exp t 1 t 1 exp t 1. Proof It follows from Theorem. that t T Z t K exp t T Z α t K exp ] 1α ds t T 1 K/Z t exp t T 1 K/Z α t exp ] f s ds 1α ds. Thus, the pricing formula of American currency call option follows from Definition 4.3 formulas 7, 8 9 immediately. Theorem 4.6 Let C A be the American currency call option price under model 6. Then, 1 C A is a decreasing function of m 1 ; C A is a decreasing function of m ; 3 C A is an increasing function of μ; 4 C A is a decreasing function of r ; 5 C A is a decreasing function of f ; 6 C A is an increasing function of Z ; 7 C A is a decreasing function of K. Proof By Theorem 4.5, the pricing formula of American currency call option can be expressed as C A 1 t T Z expμt σ t 1 α K exp m r 1t m 1 exp t 1 σ 1 1 1 α 1 exp t T t 1 exp t 1 Z m t σ 1 1 α K expμt σ t 1 α f m exp t 1 t 1 exp t 1. 1 Since t exp t1 >, t >, the function exp m 1t m 1 a1 exp t 1 exp m 1 a1 t exp t 1 is decreasing with respect to m 1 the American currency call option price C A is decreasing with respect to the parameter m 1.

An uncertain currency model with floating interest rates 6747 Since t exp t1 >, t >, the function exp m t m a exp t 1 exp m a t exp t 1 is decreasing with respect to m the American currency call option price C A is decreasing with respect to the parameter m. 3 Since the functions t T Z exp μt σ t 1 α K Z K/ expμt σ t 1 α K, Z, T > t T are increasing with respect to μ, the American currency call option price C A is increasing with respect to the log-drift μ of exchange rate. 4 Since the function expr is decreasing with respect to r, the American currency call option price C A is decreasing with respect to the initial domestic interest rate r. 5 Since the function exp f is decreasing with respect to f, the American currency call option price C A is decreasing with respect to the initial foreign interest rate f. 6 Since the functions Z exp μt σ t 1 α K t T Z K/ expμt σ t 1 α K, Z, T > t T are increasing with respect to Z, the American currency call option price C A is increasing with respect to the initial exchange rate Z. 7 Obviously, the functions t T Z exp μt σ t 1 α K Z K/ expμt σ t 1 α K, Z, T > t T are decreasing with respect to K. Therefore, the American currency call option price C A is decreasing with respect to the strike price K. By following the same method as American currency call option, we obtain the following definition, formula properties of American currency put option pricing with strike price K maturity at time T. Definition 4.4 Assume an American currency put option has a strike price K a maturity date T. Then, the American currency put option price is P A 1 exp t T 1 Z exp t T ] K Z t ] f s ds K/Z t 1. Theorem 4.7 Assume an American currency put option under model 6 has a strike price K a maturity date T. Then, the American currency put option price is P A 1 Z t T K Zt α exp t T K/Z α t 1 exp r α s ds f α s ds where K Zt α K Z expμt σ t α 1, K/Zt α 1 K Z expμt σ t 1 α 1, exp α ds exp m r 1t m 1 exp t 1 σ 1 1 α exp exp t 1 exp t 1 f α s ds m t σ 1 α f m exp t 1 t 1 exp t 1.

6748 X. Wang, Y. Ning Proof It follows from Theorem. that t T t T K Z t exp t T K Z α t exp K/Z t 1 exp t T ] α ds K/Z α t 1 exp ] f s ds f α s ds. Thus, the pricing formula of American currency put option follows from Definition 4.4 formulas 7, 8 9 immediately. Theorem 4.8 Let P A be the American currency put option price under model 6. Then, 1 P A is a decreasing function of m 1 ; P A is a decreasing function of m ; 3 P A is a decreasing function of μ; 4 P A is a decreasing function of r ; 5 P A is a decreasing function of f ; 6 P A is a decreasing function of Z ; 7 P A is an increasing function of K. Proof By Theorem 4.7, the pricing formula of American currency put option can be expressed as P A 1 t T K Z expμt σ t 1 α exp m r 1t m 1 exp t 1 σ 1 1 α 1 exp t T m t σ 1 α t 1 exp t 1 K expμt σ t 1 α Z f m exp t 1 t 1 exp t 1. 1 Since t exp t1 >, t >, the function exp m 1t m 1 a1 exp t 1 exp m 1 a1 t exp t 1 is decreasing with respect to m 1 the American currency put option price P A is decreasing with respect to the parameter m 1. Since t exp t1 >, t >, the function exp m t m a exp t 1 exp m a t exp t 1 is decreasing with respect to m the American currency put option price P A is decreasing with respect to the parameter m. 3 Since the functions K Z exp t T μt σ t 1 α K/ expμt σ t 1 α Z K, Z, T > t T are decreasing with respect to μ, the American currency put option price P A is decreasing with respect to the log-drift μ of exchange rate. 4 Since the function expr is decreasing with respect to r, the American currency put option price P A is decreasing with respect to the initial domestic interest rate r. 5 Since the function exp f is decreasing with respect to f, the American currency put option price P A is decreasing with respect to the initial foreign interest rate f. 6 Since the functions K Z exp t T μt σ t 1 α K/ expμt σ t 1 α Z K, Z, T > t T are decreasing with respect to Z, the American currency put option price P A is decreasing with respect to the initial exchange rate Z.

An uncertain currency model with floating interest rates 6749 Table 1 Paramete setting in G K model, Liu Chen Ralescu model model 6 G K Liu Chen Ralescu Model 6 G K Liu Chen Ralescu Model 6 r% 1.1 1.1 σ 1. f %.44.44 σ.3 m 1 %.11 σ.1.1.1 m %.44 μ.65.65.65.1 r % 1. 1. 1..1 f %.3.3.3 Z 6.58 6.58 6.58 K 6.3 6.3 6.3 7 Obviously, the functions t T K Z exp μt σ t 1 α K/ expμt σ t 1 α Z K, Z, T > t T are increasing with respect to K. Therefore, the American currency put option price P A is increasing with respect to the strike price K. 5 Numerical experiments In Section 4, we have derived the pricing formulas of uropean American currency option analyzed the relationship between the pricing formulas of currency option seven paramete m 1, m,μ,r, f, Z, K. Now, the aim of this section is to show the performance of our proposed model 6 to present other paramete impact on the pricing formulas under model 6. For these purposes, we carry on a series of experiments. These experiments are not based on empirical data but artificial data. Moreover, for the sake of simplicity, we just consider the uropean currency call option case. Indeed, using the same method, one can also discuss the remaining three cases. 5.1 Comparative analysis To show the preference of our proposed model, we compute the uropean currency call option price using model 6 make comparisons with the result of the G K model Liu Chen Ralescu model. Table 1 presents the paramete for computing the hypothetical uropean currency call option price, where r r m 1 / f f m / represent the domestic interest rate foreign interest rate in the G K model Liu Chen Ralescu model, respectively. Fig. 1 uropean currency call option price C via G K model, Liu Chen Ralescu model model 6 Remark 5.1 Assume that a uropean currency call option follows the G K model has a strike price K a maturitydate T. Garman Kohlhagen give the pricing formula of uropean currency call option as follows Garman Kohlhagen 1983: P Z exp ftnϑ K exprtnϑ σ T where ϑ lnz /K r f σ / T σ T N is the cumulative normal distribution function. Under the assumption that the exchange rate, respectively, follows G K model, Liu Chen Ralescu model model 6, we compute the prices of uropean currency call option computed by these three models depict them in Fig. 1, where the horizontal axis is the maturity date T the vertical axis is the uropean currency call option price C. Figure 1 shows the difference of uropean currency call option prices via the three approaches is very little when the maturity date T, 5]. In addition, we can also investigate that the prices obtained by model 6 are larger than the prices

675 X. Wang, Y. Ning Fig. uropean currency call option price C veus different values of the paramete σ 1, σ σ,wherea shows the relationship between C σ 1 ; b shows the relationship between C σ ; c shows the relationship between C σ Fig. 3 uropean currency call option price C veus different values of the paramete,wherea shows the relationship between C ; b shows the relationship between C obtained by the Liu Chen Ralescu model. The main reason is that both domestic interest rate foreign interest rate are floating. Next, we want to investigate whether they are close to hypothetical option prices. According to our intuition common sense, the price of uropean currency call option should increase as the maturity date increases. The price computed by model 6 suits this principle. However, the prices computed by G K model Liu Chen Ralescu model do not suit this principle because they both increase fit then decrease as the maturity date goes infinity. Hence, model 6 seems more reasonable to compute the uropean currency call option price compared to G K model Liu Chen Ralescu model. 5. Parametric analysis In what follows, we will study the influence of the paramete,,σ 1,σ,σ,T on the pricing formula of uropean currency call option C present the results of a series of experiments. At fit, we consider one case where the paramete,, T are fixed, the other paramete σ 1,σ,σ are changed, then consider the prices for different paramete. Finally, we study the relationship between T C when the other paramete remain unchanged. Figures, 3 4 display the prices of a uropean currency call option veus its paramete,,σ 1,σ,σ T.The defaulting paramete are shown in Table 1, the maturity date T 1.

An uncertain currency model with floating interest rates 6751 C 1 Z α T K exp 1α ds Z 1 K/ZT α exp or P 1 K ZT α exp Z K/Z α T 1 exp Proof By Theorem., we can obtain 1α ds f α s ds r α s ds. Fig. 4 uropean currency call option price C veus different values of the parameter T Figures, 3 4 indicate that the uropean currency call option price C is increasing with respect to the paramete σ 1,σ, σ,, the maturity date T. 6 Currency option pricing under general currency model In this section, we continue to derive the pricing formulas of currency option based on the assumption that domestic interest rate, foreign interest rate the exchange rate follow a general uncertain differential equation, respectively. Assume that the domestic interest rate, the foreign interest rate the exchange rate follow general uncertain differential equations. Then, a general currency model can be expressed as follows, dr t F 1 t, r t dt G 1 t, r t dc 1t d f t F t, f t dt G t, f t dc t 1 dz t Ft, Z t dt Gt, Z t dc 3t where F 1, F, F, G 1, G G are six real functions C 1t, C t C 3t are independent canonical Liu processes. Remark 6.1 The invee uncertainty distributions of the domestic interest rate r t, foreign interest rate f t exchange rate Z t are their α-path rt α, f t α Zt α, respectively, which can be calculated by some numerical methods, such as Adams Simpson method Wang et al. 15. Theorem 6.1 Assume a uropean currency call or put option under model 1 has a strike price K a maturity date T. Then, the uropean currency call or put option price is Z T K exp Z α T K exp ] 1 K/Z T exp 1 K/Z α T exp 1α ds ] f s ds 1α ds. Thus, the pricing formula of uropean currency call option follows from Definition 4.1 immediately. Theorem 6. Assume an American currency call or put option under model 1 has a strike price K a maturity date T. Then, the American currency call or put option price is C A 1 t T Z or P A 1 Z Zt α K exp t T 1 K/Z α t exp t T t T 1α ds K Z α t exp K/Z α t 1 exp Proof It follows from Theorem. that t T Z t K exp t T Z α t K exp ] 1α ds. 1α ds r α s ds α ds.

675 X. Wang, Y. Ning t T 1 K/Z t exp t T 1 K/Z α t exp ] f s ds 1α ds. Thus, the pricing formula of American currency call option follows from Definition 4.3 immediately. 7 Conclusions In this paper, we proposed an uncertain currency model under uncertain interest rates. Different from Liu Chen Ralescu model, both the domestic interest rate foreign interest rate in this model follow uncertain differential equations instead of constants. Subsequently, the pricing formulas of uropean American currency option under the new currency model were derived in the form of invee uncertainty distribution. Furthermore, the pricing formulas of uropean currency call option are increasing with respect to the initial exchange rate Z, the log-drift μ of exchange rate, the diffusion σ 1,σ,the log-diffusion σ of exchange rate, the paramete, the maturity date T, while they are decreasing with respect to the initial domestic interest rate r, the initial foreign interest rate f, the strike price K the paramete m 1, m. Some numerical experiments were designed to show that our proposed currency model is more reasonable to compute the currency option price compared to other currency models. Finally, we proposed a more general currency model derived the pricing formulas of uropean American currency option under this model. Acknowledgements This work is ported by Natural Science Foundation of Shong Province ZR14GL. Compliance with ethical stards Conflict of interest The autho declare that they have no conflict of interest to this work. Appendix 1: Uncertain variable Definition 7.1 Liu 7, 9 Let L be a σ -algebra on a nonempty set Ɣ. AsetfunctionM : L, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 1 Normality Axiom M{Ɣ} 1; Axiom Duality Axiom M{ }M{ c }1 for any event L; Axiom 3 Subadditivity Axiom For every countable sequence of { i } L, wehave { } M i M{ i }. i1 k1 i1 Axiom 4 Product Axiom LetƔ k, L k, M k be uncertainty spaces for k 1,,...The product uncertain measure M is an uncertain measure satisfying { } M k M k { k } k1 where a b mina, b k L k for k 1,,..., respectively. Definition 7. Liu 7An uncertain variable ξ is a measurable function from an uncertainty space Ɣ, L, M to the set of real numbe, i.e., for any Borel set B of real numbe, the set {ξ B} {γ Ɣ ξγ B} is an event. To describe an uncertain variable ξ in practice, a concept of uncertainty distribution was defined by Liu 7as x M {ξ x} for any real number x.if has an invee function, 1 is called the invee uncertainty distribution of ξ Liu 1. Definition 7.3 Liu 9 Uncertain variables ξ 1,ξ,...,ξ n aresaidtobeindependent if { n } M ξ i B i i1 n M {ξ i B i } i1 for any Borel sets B 1, B,...,B n of real numbe. Definition 7.4 Liu 7 Letξ be an uncertain variable. The expected value of ξ is defined by ξ] M{ξ r}dr M{ξ r}dr provided that at least one of the above two integrals is finite. Theorem 7.1 Liu 1 Assume ξ 1,ξ,...,ξ n are independent uncertain variables with regular uncertainty distributions 1,,..., n, respectively. If the function f x 1, x,...,x n is strictly increasing with respect to x 1, x,...,x m strictly decreasing with respect to x m1, x m,...,x n,

An uncertain currency model with floating interest rates 6753 then ξ f ξ 1,ξ,...,ξ n has an invee uncertainty distribution 1 α f 1 1 α,..., 1 m α, 1 m1 1 α,..., 1 n 1 α. In addition, Liu H1 proved that the uncertain variable ξ has an expected value ξ] f 1 1 α,..., 1 m α, 1 m1 1 α,..., 1 n 1 α. Appendix : Uncertain process An uncertain process is essentially a sequence of uncertain variables indexed by time or space defined as follows: Definition 7.5 Liu 8 LetƔ,L, M be an uncertainty space T be a totally ordered set e.g., time. An uncertain process is a function X t γ from T Ɣ, L, M to the set of real numbe such that {X t B} {γ Ɣ X t γ B} is an event for any Borel set B at each t. For each γ Ɣ,the function X t γ is called a sample path of X t. Definition 7.6 Liu 8 An uncertain process X t is said to have an uncertainty distribution t x if at each t, the uncertain variable X t has the uncertainty distribution t x. And the invee function 1 t α of t x is called the invee uncertainty distribution of X t. Definition 7.7 Liu 14 Uncertain processes X 1t, X t,..., X nt are said to be independent if for any positive integer k any t 1, t,...,t k, the uncertain vecto ξ i X it1, X it,...,x itk, i 1,,...,n are independent, i.e., for any k-dimensional Borel sets B 1, B,...,B n,wehave { n } M ξ i B i i1 n M{ξ i B i }. i1 Theorem 7. Liu 14 Assume X 1t, X t,...,x nt are independent uncertain processes with regular uncertainty distributions 1t, t,..., nt, respectively. If the function f x 1, x,...,x n is strictly increasing with respect to x 1, x,...,x m strictly decreasing with respect to x m1, x m,...,x n, uncertain process X t f X 1t, X t,..., X nt has an invee uncertainty distribution t 1 α f 1 1t α,..., 1 mt α, 1 m1,t 1 α,..., 1 nt 1 α. Definition 7.8 Liu 9 An uncertain process C t t is called a canonical Liu process if i C almost all sample paths are Lipschitz continuous, ii C t is a stationary independent increment process, iii every increment C st C s is a normal uncertain variable with expected value variance t. Definition 7.9 Liu 9LetC t be a canonical Liu process. Then, for any real numbe μ σ >, the uncertain process G t expμt σ C t is called a geometric Liu process, where μ is called the logdrift σ is called the log-diffusion. Definition 7.1 Liu 9 LetX t be an uncertain process let C t be a canonical Liu process. For any partition of the closed interval a, b] with a t 1 < t < < t k1 b, the mesh is written as t i1 t i. 1 i k Then, the Liu integral of X t with respect to C t is defined by b a X t dc t lim i1 k X ti C ti1 C ti provided that the limit exists almost surely is finite. In this case, the uncertain process X t is said to be integrable. References Amin K, Jarrow R 1991 Pricing foreign currency options under stochastic interest rates. J Int Money Finance 13:31 39 Black F, Scholes M 1973 The pricing of options corporate liabilities. J Polit con 811:637 654 Bollen N, Rasiel 3 The performance of alternative valuation models in the OTC currency options market. J Int Money Finance 1:33 64 Carr P, Wu L 7 Stochastic skew in currency options. J Financ con 861:13 47 Chen X, Gao J 13 Uncertain term structure model of interest rate. Soft Comput 174:597 64

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